Searcharxiv⌕ Search

arXiv subjects

R. Oset Sinha

Publications and source records attributed to R. Oset Sinha.

9 recordsLinked to original sources
arXiv2026

Recovery problem of parametrizations from Legendre data

The problem of recovery of parametrizations from Legendre data is a very important inverse problem. In this paper, we provide a systematic and widely-applicable method to recover parametrizations $f: U_n \to \mathbb{R}^{n+1}$ from Legendre data where $U_n$ is an open subset of $\mathbb R^n$. Namely, for a dense subset of the space of real-analytic parametrizations from $U_n$ into $\mathbb{R}^{n+1}$, we show how to recover the parametrization from the Gauss mapping and the height function. Moreover, in order to assist readers to apply results of this paper, many concrete examples are given.

math.DG↗
arXiv2025

Counting $D_4$ singularities in the image of a wave front

We give a formula to count the number of $D_4$ singularities in a stable frontal perturbation of a corank $2$ wave front singularity $f\colon (\mathbb{C}^3,0) \to (\mathbb{C}^4,0)$ using Mond's method of stable perturbations of map germs. For a generic germ of corank $2$ wave front $f\colon (\mathbb{C}^3,S) \to (\mathbb{C}^4,0)$, the image of a stable deformation $f_t$ of $f$ exhibits $A_k$ singularities with $k \leq 4$, their transverse intersections and the aforementioned $D_4$ singularities for $0 < |t| \ll 1$. By interpreting the image of $f_t$ as the discriminant (the image of the critical point set) of a smooth map germ $H_t\colon (\mathbb{C}^5,0) \to (\mathbb{C}^4,0)$, we define an algebra whose dimension over $\mathbb{C}$ is equal to the number of $D_4$ points in the image of $f_t$.

math.AG↗
arXiv2024

A proof of the Mond conjecture for wave fronts

We prove the Mond conjecture for wave fronts which states that the number of parameters of a frontal versal unfolding is less than or equal to the number of spheres in the image of a stable frontal deformation with equality if the wave front is weighted homogeneous. We give two different proofs. The first one depends on the fact that wave fronts are related to discriminants of map germs and we then use the analogous result proved by Damon and Mond in this context. The second one is based on ideas by Fernández de Bobadilla, Nuño-Ballesteros and Peñafort Sanchis and by Nuño-Ballesteros and Fernández-Hernández. The advantage of the second approach is that most results are valid for any frontal, not only wave fronts, and thus give important tools which may be useful to prove the conjecture for frontals in general.

math.AG↗
arXiv2023

Deformations of corank $1$ frontals

We develop a Thom-Mather theory of frontals analogous to Ishikawa's theory of deformations of Legendrian singularities but at the frontal level, avoiding the use of the contact setting. In particular, we define concepts like frontal stability, versality of frontal unfoldings or frontal codimension. We prove several characterizations of stability, including a frontal Mather-Gaffney criterion, and of versality. We then define the method of reduction with which we show how to construct frontal versal unfoldings of plane curves and show how to construct stable unfoldings of corank 1 frontals with isolated instability which are not necessarily versal. We prove a frontal version of Mond's conjecture in dimension 1. Finally, we classify stable frontal multigerms and give a complete classification of corank 1 stable frontals from $\mathbb C^3$ to $\mathbb C^4$.

math.AG↗
arXiv2022

Simplicity of augmentations of codimension 1 germs and by Morse functions

We study the simplicity of map-germs obtained by the operation of augmentation and describe how to obtain their versal unfoldings. When the augmentation comes from an $\mathscr{A}_e$-codimension 1 germ or the augmenting function is a Morse function, we give a complete characterisation for simplicity. These characterisations yield all the simple augmentations in all explicitly obtained classifications of $\mathscr{A}$-simple monogerms except for one ($F_4$ in Mond's list from $\mathbb{C}^2$ to $\mathbb{C}^3$). Moreover, using our results we produce a list of simple augmentations from $\mathbb{C}^4$ to $\mathbb{C}^4$.

math.AG↗
arXiv2022

Singularities of Frontal Surfaces

We consider singularities of frontal surfaces of corank one and finite frontal codimension. We look at the classification under left-right-equivalence and introduce the notion of frontalisation for singularities of fold type. We define the cuspidal and the transverse double point curves and prove that the frontal has finite codimension if and only if both curves are reduced. Finally, we also discuss about the frontal versions of the Marar-Mond formulas and the Mond's conjecture.

math.AG↗
arXiv2014

Liftable vector fields over corank one multigerms

In this paper, a systematic method is given to construct all liftable vector fields over an analytic multigerm $f: (\mathbb{K}^n, S)\to (\mathbb{K}^p,0)$ of corank at most one admitting a one-parameter stable unfolding.

math.AG↗
arXiv2014

Classifying codimension 2 multigerms

We generalise the operations of augmentation and concatenations in order to obtain multigerms of analytic (or smooth) maps $(\mathbb K^n,S)\rightarrow(\mathbb K^p,0)$ with $\mathbb K=\mathbb C$ or $\mathbb R$ from monogerms and some special multigerms. We then prove that any corank 1 codimension 2 multigerm in Mather's nice dimensions $(n,p)$ with $n\geq p-1$ can be constructed using augmentations and these operations.

math.CV↗